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Syllabus

Click below to expand the details for each class, including times and locations, the lecture videos you should watch in advance, the review questions you should solve in advance, and the problem sets you should complete after each class. Please note that all classes will take place at Lady Margaret Hall.

Date, Time, and Location

Thursday 29 October (MT Week 3), 3:30-6pm, Lyttleton Room, LMH

In-class demonstrations

During this class we will play two games about conditional probability: Who’s on the other side? and The lie detector. To avoid spoiling the surprise, please do not open either link before class. Bring a device that can access the internet so that you can play along.

Videos

These videos are already available on Canvas.

  • P0: Introduction
  • P1: Probability Basics
    • Probabilities as set functions; scaled-Venn representation (\(\Omega\) as the unit square)
    • Complements, intersections (“and”), unions (“or”) of events
    • Building blocks: sample space, event, event space \(\mathcal{F}\)
    • Worked sample/event spaces: fair die, coin toss, lightbulb lifetime
    • The three (Kolmogorov) axioms; disjoint events and additivity
    • The four rules: complement, union of two events, Boole/union bound, logical consequence (monotonicity), with the complement-rule derivation
  • P2: Conditional Probability
    • Conditional probability \(P(A|B)\) as information shrinking the sample space
    • Kolmogorov definition \(P(A|B)=P(A\cap B)/P(B)\); de Finetti multiplication rule
    • Venn-diagram intuition (conditioning rescales \(B\) to the unit circle)
    • Revised (conditional) probability axioms
    • Asymmetry: \(P(A|B)\neq P(B|A)\)
    • Law of Total Probability over a partition. A partition of \(\Omega\) is a collection of events that are mutually exclusive and collectively exhaustive (their union is \(\Omega\)). Exactly one of them therefore occurs. Your lecturer uses the term without defining it.
  • P3: Independence & Bayes’ Rule
    • Independence via \(P(A\cap B)=P(A)P(B)\) and \(P(A|B)=P(A)\); symmetry
    • Bayes’ Rule: statement, derivation, historical background
    • Law of Total Probability for the denominator
    • Prior / likelihood / evidence / posterior vocabulary
    • Worked example: doping/EPO test and the base-rate problem
    • Sally Clark case: independence fallacy and the prosecutor’s fallacy

Slide corrections

The slides in the lecture videos for this class contain the following errors.

  • P3, slide 10, concludes “So a failed test does mean the cyclist is guilty.” It should say “does not mean”. A failed test raises the probability of doping from the 15% base rate to only about 57%.
  • P3, slide 16, twice says that the children died “suddenly and expectedly”. It should say “suddenly and unexpectedly”, as on slides 14 and 17.

Further reading (optional)

These posts from my blog go beyond the material in the lecture videos.

Quiz

Before class, complete the review questions for class 1. These will be the basis of the quiz at the start of class.

Flashcards

Download the Class 1 flashcard deck and study it alongside the review questions. The flashcards are fair game on the quiz. See the Flashcards page for setup and study tips.

Problem Set

After attending class, complete problem set number one.

Extra Practice

For more practice, work through the Class 1 extra practice problems. They are grouped by topic and ordered by difficulty. Each problem comes with a solution.

Date, Time, and Location

Thursday 12 November (MT Week 5), 3:30-6pm, Lyttleton Room, LMH

Videos

These videos are scheduled for release on Canvas in MT Week 4 (week beginning Sunday 1 November). Start watching them as soon as they appear, since this class is in Week 5.

Before class, you are also required to read or watch the sections on joint, marginal, and conditional distributions and independence in my notes and videos, Discrete Random Variables I. They work through the material in D6 with an example. The earlier sections of that page, on the material in D2 and D4, are optional.

  • D0: Data & Variables Intro
  • D1: Data Basics
    • Variable types: categorical (binary, ordinal) vs. numerical (discrete, continuous)
    • Support of a variable; univariate vs. multivariate
    • Cross-sectional data \(\{X_i\}_{i=1,\dots,N}\); geometric depiction (number line, 2D/3D scatter)
    • Time-series data \(\{X_t\}_{t=1,\dots,T}\); a time series as one observation on a \(T\)-dimensional variable
    • Panel data; balanced vs. unbalanced
  • D2: Probability Mass Functions
    • PMF definition for discrete variables; frequentist (relative-frequency) interpretation
    • Representations: formula, table, stem (“lollipop”) plot
    • PMF properties: \(f(x)=\mathbb{P}(X=x)\ge 0\), \(\sum_x f(x)=1\); support, \(X\) vs. \(x\)
    • Bernoulli distribution; categorical/“multinoulli”; the Iverson bracket \([x=y]\), which is 1 if \(x=y\) and 0 otherwise (this is your lecturer’s notation; outside this course, the more common notation is the indicator function \(\mathbf{1}\{x=y\}\))
    • Summation (\(\Sigma\)) and product (\(\Pi\)) notation
    • Binomial distribution: PMF, parameters \(N\) and \(p\), shape
  • D3: Probability Density Functions
    • PDF as a limit; the continuous analogue of the PMF (smoothed histogram)
    • Properties: \(f(x)\ge 0\) and \(\int f(x)\,dx=1\)
    • Subtleties: \(\mathbb{P}(X=x)=0\); \(f(x)\) is not a probability
    • Continuous Uniform: \(f(x)=1/(b-a)\)
    • Normal (Gaussian): density; \(\mu\) controls location, \(\sigma\) spread
    • Probabilities defined over intervals, not points
  • D4: Cumulative Distribution Functions
    • CDF \(F(x)=\mathbb{P}(X\le x)\); right-continuity
    • Discrete CDFs as step functions; continuous CDFs (Uniform, Normal S-curve)
    • Properties: bounds, monotonicity, \(\mathbb{P}(X>x)=1-F(x)\), \(F(b)-F(a)\)
    • PDF–CDF link: \(f=dF/dx\), \(F=\int_{-\infty}^x f\) (FTC)
    • Quantiles from the CDF; reading Standard Normal tables with interpolation
    • Key critical values: \(\pm1.645\) (90%), \(\pm1.96\) (95%), \(\pm2.58\) (99%)
  • D5: Standardization
    • Standard Normal \(N(0,1)\) and the need to standardize general \(N(\mu,\sigma^2)\)
    • Adding a constant (location shift) vs. multiplying (scales location and spread)
    • Standardization \(Z=(X-\mu)/\sigma\sim N(0,1)\), derived in two steps
    • Computing \(\mathbb{P}(X\le x)\) by standardizing; left-tail, right-tail, interval examples
    • Quantiles: look up the \(N(0,1)\) quantile, then de-standardize
  • D6: Multivariate Distributions
    • Bivariate/multivariate data; joint CDF and joint PDF/PMF
    • Marginal distributions from the joint
    • Independence: joint factors into the product of marginals
    • Conditional distributions as scaled slices through the joint
    • Generalization to higher dimensions

Slide corrections

The slides in the lecture videos for this class contain the following errors.

  • D2, slide 12, defines the Bernoulli pmf as \(f(0) = \mathbb{P}(X = 0) = p\) and \(f(1) = \mathbb{P}(X = 1) = 1 - p\). The usual convention is the reverse: \(f(1) = \mathbb{P}(X = 1) = p\) and \(f(0) = \mathbb{P}(X = 0) = 1 - p\), so that \(p\) is the probability of a one. Slide 19 also uses this convention.
  • D2, slide 19, says that the scaling factor \(K\) in \(f(x) = K p^x (1-p)^{N-x}\) depends on \(N\) and \(p\). In fact \(K = \binom{N}{x} = \frac{N!}{x!(N-x)!}\), the number of different strings with \(x\) ones and \(N - x\) zeros. It depends on \(N\) and \(x\) but not on \(p\). Each such string has probability \(p^x(1-p)^{N-x}\), so the probability that the sum \(Y\) from slide 18 equals \(x\) is \(\binom{N}{x} p^x (1-p)^{N-x}\). The slide writes \(X = x\) for this event, where it means \(Y = x\).
  • D4, slide 27, says that \(\mathbb{P}(X \le -2.33) = 0.099\). The value given in the line above, and in the table, is 0.0099.
  • D4, slide 32, item 5, should read: the values \(\pm x\) such that \(\mathbb{P}(|X| \le x) = 0.95\) are \(\pm 1.96\). The value 2.5758 corresponds to \(\mathbb{P}(|X| \le x) = 0.99\).

Quiz

Before class, complete the review questions for class 2. These will be the basis of the quiz at the start of class. There will also be some additional questions based on the material covered in class 1.

Flashcards

Download the Class 2 flashcard deck and study it alongside the review questions. The flashcards are fair game on the quiz. See the Flashcards page for setup and study tips.

Problem Set

After attending class, complete problem set number two.

Date, Time, and Location

Thursday 26 November (MT Week 7), 3:30-6pm, Lyttleton Room, LMH

Videos

These videos are scheduled for release on Canvas in MT Week 4 (week beginning Sunday 1 November), together with the videos for Class 2.

Before class, you are also required to read or watch my notes and videos, Discrete Random Variables II. They work through the material in M1 to M4 with examples.

  • M0: Intermission
  • M1: Expected Values
    • Expected value (discrete sum vs. continuous integral) as a probability-weighted average
    • Mean need not lie in the support or be the most probable value
    • Means of Uniform \((a+b)/2\) and Normal \(\mu\); expectation of a binary variable equals \(p\)
    • \(\mathbb{E}[g(X)]\) for a function of a variable
    • Rules: affine transformations, additivity (linearity), multiplicativity under independence
    • Jensen’s inequality (concave vs. convex); the heavy-tailed Cauchy (no mean)
  • M2: Variance
    • Variance \(=\mathbb{E}([X-\mathbb{E}X]^2)=\mathbb{E}(X^2)-[\mathbb{E}X]^2\); non-negativity
    • Discrete vs. continuous formulas; standard deviation and units
    • Variance of Uniform \((b-a)^2/12\), Normal \(\sigma^2\), binary \(p(1-p)\)
    • Standardization \(Z=(X-\mu)/\sigma\sim N(0,1)\)
    • The empirical (68–95–99.7) rule
  • M3: Conditional Expectations
    • Conditional expectation \(\mathbb{E}(Y|X)\) as the mean of a conditional distribution
    • Conditional means for prediction; gender wage gap as \(\mathbb{E}[Y|X=1]-\mathbb{E}[Y|X=0]\)
    • Formal definition (discrete and continuous)
    • Properties: pull-out rule, independence, Law of Iterated Expectations
    • \(\mathbb{E}[Y|X=x]\) (a point) vs. \(\mathbb{E}[Y|X]\) (a function)
  • M4: Correlation, Covariance, & Independence
    • Covariance: definition, sign interpretation, scatter-plot intuition
    • Properties: symmetry, \(\text{Cov}(X,X)=\text{Var}(X)\), \(\text{Cov}(X,a)=0\), bilinearity
    • Variance of sums \(\text{Var}(aX\pm bY)=a^2\text{Var}(X)+b^2\text{Var}(Y)\pm2ab\,\text{Cov}(X,Y)\)
    • Independence \(\Rightarrow\) zero covariance (converse fails)
    • Correlation as covariance of standardized variables
    • Mean-independence as an intermediate concept

Further reading (optional)

These posts from my blog go beyond the material in the lecture videos.

  • The Independence Zoo distinguishes independence, mean independence, and zero correlation. It includes examples of random variables that are dependent but uncorrelated.
  • Random Variables Cheatsheet points to my reference sheet of the main rules for expected value, variance, and covariance.

Quiz

Before class, complete the review questions for class 3. These will be the basis of the quiz at the start of class. There will also be some additional questions based on the material covered in class 2.

Flashcards

Download the Class 3 flashcard deck and study it alongside the review questions. The flashcards are fair game on the quiz. See the Flashcards page for setup and study tips.

Problem Set

After attending class, complete problem set number three. Over the vacation make sure to study for your collection on probability! This will take place in HT Week 0.

Date, Time, and Location

Thursday 28 January (HT Week 2), 3:30-6pm, Lyttleton Room, LMH

Videos

These videos are scheduled for release on Canvas in MT Week 8 (week beginning Sunday 29 November).

  • S0: Statistics Intro
  • S1: Populations, Samples & Random Variables
    • Populations (enumeration vs. model); samples and representativeness
    • Sampling frames; simple random sampling vs. quota sampling; non-response/selection bias
    • Random variables as functions of uncertain events; \(X_1\) inherits the population distribution
    • iid; sampling with vs. without replacement; large-population approximation
    • Moments of iid samples equal population moments
    • Your lecturer says that random variables are “generated by random sampling,” and he calls height in the population a “variable” and height in a random sample a “random variable.” Random sampling is one way random variables arise, but there are others: the outcome of a die roll is also a random variable, and no population is involved. Outside this course, the population distribution of height is usually described as the distribution of a random variable, the height of one randomly chosen person.
  • S2: Parameters, Statistics & Estimation
    • Parameters vs. statistics; sampling variation
    • Estimand / estimator / estimate; notation (\(\hat\theta\), \(\bar X\), \(s^2\), \(\hat p\))
    • The analogue approach; standard estimators (mean, variance, covariance, correlation)
    • Sampling distribution of an estimator
    • Unbiasedness; bias of the sample variance and Bessel’s correction
    • Efficiency (\(\sigma^2/N\)); consistency; consistent-but-biased estimators
  • S3: The Law of Large Numbers & Central Limit Theorem
    • The single-sample inference problem; sampling distribution of the mean
    • LLN (Bernoulli 1713): \(\bar X_N \to \mu\) in probability
    • CLT (Laplace 1810): standardized mean \(\to N(0,1)\)
    • Implied approximation \(\bar X_N \sim N(\mu,\sigma^2/N)\)
    • Standardization and the location-scale transformation
    • Simulation illustration; LLN/CLT as foundations of inference
  • S4: Intro to Statistical Inference
    • Inference: recovering population parameters from sample data
    • Estimates vs. parameters; the sampling distribution
    • The two tools: confidence intervals and hypothesis tests
    • Parameter/estimator table (general vs. binary)
    • CLT-based normal approximation; plugging in \(s^2\) for \(\sigma^2\); nuisance parameters
  • S5: Confidence Intervals Overview
    • Deriving a CI for the mean from an iid sample
    • Standardizing and invoking the CLT; the 95% probability statement
    • General form: statistic \(\pm\) (number of std. errors) \(\times\) std. error; critical values
    • CI width vs. confidence level, std. error, variance, sample size
    • The CI as a statistic; correct vs. incorrect (frequentist) interpretation
  • S5a: Confidence Intervals for Means
    • CI for a single mean: \(\bar X \pm 1.96\sqrt{s^2/N}\)
    • Standard error of the mean; effect of \(N\) and variance on width
    • CI for the difference of two independent means
    • Paired (dependent) samples: \(\text{Var}(\bar X-\bar Y)=\text{Var}(\bar D)\)
    • Worked examples (anchoring experiment; paired exams)
  • S5b: Confidence Intervals for Proportions
    • CI for a single proportion: \(\hat p \pm 1.96\sqrt{\hat p(1-\hat p)/N}\)
    • CI for the difference of two independent proportions
    • Paired (dependent) proportions as a single sample of differences
    • Why paired binary differences are “trinary” \(\{-1,0,+1\}\)
    • Worked examples (Trump approval; partisan; smoking cessation)

Slide corrections

The slides in the lecture videos for this class contain the following errors.

  • S1, slide 19, refers to “a population of 1000 scripts instead of 100”. The example on slide 16 had a population of 10.
  • S2, slide 8, gives the covariance of two binary variables as \(\sigma_{xy} = p_x p_y\). This is \(\mathbb{E}[X]\,\mathbb{E}[Y]\), not the covariance. The covariance is \(\sigma_{xy} = p_{xy} - p_x p_y\), where \(p_{xy} = \mathbb{P}(X = 1, Y = 1)\). It equals zero when \(X\) and \(Y\) are independent. The corresponding sample covariance is \(s_{xy} = \frac{N}{N-1}(\hat{p}_{xy} - \hat{p}_x \hat{p}_y)\), where \(\hat{p}_{xy}\) is the sample proportion with \(X_i = Y_i = 1\). The slide writes it as \(s_{xy}^2 = \hat{p}_x \hat{p}_y\).
  • S5, slide 9, and S5b, slide 5, say that a computed confidence interval has a 95% probability of covering the parameter. A particular computed interval, such as \([0.3653, 0.4147]\) on S5b slide 5, is a realization: it either covers the parameter or it doesn’t. The 95% describes the procedure that produced it: if you repeated that procedure many times, 95% of the intervals you constructed would cover the true parameter. Read the “RIGHT” statement on S5 slide 10 in this way, with \([a, b]\) standing for the random interval before the data are drawn.
  • S5a, slide 9, labels the “Hi” group’s variance \(s_x^2 = 253\). The “Hi” group is the \(Y\) sample, so the label should be \(s_y^2\). The calculation on slide 10 uses the value correctly.
  • S5b, slide 10, gives a sample correlation of 0.66 and a variance of the differences of 0.4. These are inconsistent with the two sample variances, 0.21 and 0.1875. With a correlation of 0.66, the variance of the differences is \(0.21 + 0.1875 - 2(0.66)\sqrt{0.21 \times 0.1875} \approx 0.136\). Using 0.136, the interval on slide 11 becomes \(-0.05 \pm 1.96\sqrt{0.136/100} = [-0.122, 0.022]\) instead of \([-0.174, 0.074]\). It still covers zero, so the conclusion does not change.

Further reading (optional)

These posts from my blog go beyond the material in the lecture videos.

The next two posts concern the confidence interval for a proportion. The standard interval from lecture video S5b is the one used in this course and its exams. These posts explain its weaknesses and describe a better alternative.

Quiz

Before class, complete the review questions for class 4. These will be the basis of the quiz at the start of class. (Since you have a collection on probability in HT Week 0, there will be no “extra challenge” questions based on Class 3 on this quiz.)

Flashcards

Download the Class 4 flashcard deck and study it alongside the review questions. The flashcards are fair game on the quiz. See the Flashcards page for setup and study tips.

Problem Set

After attending class, complete problem set number four.

Date, Time, and Location

Thursday 11 February (HT Week 4), 3:30-6pm, Lyttleton Room, LMH

Videos

These videos are scheduled for release on Canvas in MT Week 8 (week beginning Sunday 29 November), together with the videos for Class 4.

  • S6: Tests of Statistical Hypotheses - Overview
    • Null vs. alternative hypotheses (about parameters, not statistics)
    • Type 1 and Type 2 errors; criminal-trial analogy
    • The five-step testing procedure
    • \(Z\) statistic \(=\) (statistic \(-\) hypothesized) / std. error; \(N(0,1)\) null
    • Critical values, \(\alpha\), confidence level; \(|Z|>c\) rule
    • \(\alpha\) as long-run Type 1 error rate; the replication crisis
  • S6a: Tests for Means
    • General test statistic form; the 5-step procedure for means
    • One-sample test for a single mean
    • Two-sample test for a difference in independent means; SE \(=\sqrt{s_x^2/N_x+s_y^2/N_y}\)
    • Paired test reduced to a one-sample test on differences
    • Worked examples (single mean; anchoring; paired exams)
  • S6b: Tests for Proportions
    • Two complications: variance derivable from the proportion; SE under the null
    • Single-proportion test using \(p_0\) for the exact SE
    • Two independent proportions: pooled estimator \(\hat\pi\) and its SE
    • Paired-proportions test on the difference variable
    • Worked examples (Trump approval; partisan; smoking cessation)
  • S7: Confidence Intervals and Hypothesis Tests Revisited
    • Duality between confidence intervals and hypothesis tests
    • The CI as the set of null values not rejected by the test
    • Two-sided z-test; finding the boundary null values
    • Deriving the 95% CI from the non-rejection region

Slide corrections

The slides in the lecture videos for this class contain the following errors.

  • S6, slide 9, says “If it is very far from one then what we have seen is extremely unlikely”. It should say “very far from 0”, the value around which \(Z\) is centered under the null.
  • S6a, slide 10, labels the “Hi” group’s variance \(s_x^2 = 253\). It should be \(s_y^2\), since the “Hi” group is the \(Y\) sample. The calculation on slide 11 uses the value correctly.
  • S6b, slide 12, writes the numerator of the test statistic as \((0.47 - 0.39)\). It should be \((0.47 - 0.31)\). The reported value \(z = 3.5819\) is correct.
  • S6b, slides 15 and 16, use the same smoking example as S5b slide 10. Its correlation of 0.66 and variance of the differences of 0.4 are inconsistent (see the Class 4 corrections). With the corrected variance of about 0.136, the test statistic on slide 16 is about 1.36 rather than 0.7906. The null is still not rejected at the 5% level. Slide 16 also writes the numerator as \((\hat{p}_x - \hat{p}_x)\), where it means \((\hat{p}_x - \hat{p}_y)\).
  • S7, slide 8, gives the test statistic as \(Z = -1.97\). The numerator is \(+1.97\sqrt{s^2/N}\), so \(Z = +1.97\). The decision to reject is unchanged.

Further reading (optional)

These posts from my blog go beyond the material in the lecture videos.

Quiz

Before class, complete the review questions for class 5. These will be the basis of the quiz at the start of class. There will also be some additional questions based on the material covered in class 4.

Flashcards

Download the Class 5 flashcard deck and study it alongside the review questions. The flashcards are fair game on the quiz. See the Flashcards page for setup and study tips.

Problem Set

After attending class, complete problem set number five.

Date, Time, and Location

Thursday 25 February (HT Week 6), 3:30-6pm, Lyttleton Room, LMH

Videos

These videos are scheduled for release on Canvas in HT Week 1 (week beginning Sunday 17 January).

  • T0: Time Series Introduction
  • T1: Time Series Data
    • Time series: chronologically ordered, serially dependent observations; modeling and forecasting
    • Trends and log transformations; breaks; seasonality; cycles; noise
    • Correlation, lags (\(X_{t-h}\)) and leads (\(X_{t+h}\))
    • Autocorrelation \(\rho(h)\); bounds and \(\rho(0)=1\); less reliable at large lags
    • The correlogram
  • T2: Basic Operations
    • Notation \(X_t\); lags, leads, initial condition \(X_0\)
    • First difference \(\Delta X_t=X_t-X_{t-1}\); seasonal (fourth) difference
    • Per-period growth rate \(g\)
    • Compound average growth rate over \(h\) periods (not the arithmetic mean of growth rates)
  • T3: Deterministic & Stochastic Time Series
    • Deterministic vs. stochastic components
    • Linear trend \(X_t=\beta_0+\beta_1 t\); aperiodic deterministic signal (irrational frequency ratio)
    • Stochastic series; iid Gaussian noise as “as-if random”
    • Structural break (regime shift); stochastic trend (time-varying mean); changing volatility
  • T4: Stationarity
    • Stationarity as the time-series analogue of iid
    • Strong vs. weak stationarity (constant mean/variance; autocovariance depends only on lag \(h\))
    • Characteristics of a stationary series
    • Inducing stationarity: differencing, detrending, variance-stabilizing transforms
    • Detecting non-stationarity (formal tests vs. “eye-ball econometrics”)
  • T5: Some Time Series Models
    • White noise (iid, zero mean, serially uncorrelated); Gaussian white noise as “shocks”
    • Random walk \(X_t=X_{t-1}+e_t\); non-stationarity (\(\text{Var}\) grows linearly in \(t\)); high persistence
    • Random walk with drift
    • AR(\(p\)) and AR(1) \(X_t=\theta_0+\theta_1 X_{t-1}+e_t\), nesting white noise / random walk / RW-with-drift; role of \(\theta_1\)’s sign
  • T6: Mean Reversion
    • Stationarity of AR(1): constant mean \(\theta_0/(1-\theta_1)\) and variance \(\sigma^2/(1-\theta_1^2)\)
    • Stationarity condition \(|\theta_1|<1\)
    • Autocorrelation \(\theta_1^h\) (exponential decay; sign alternation when \(\theta_1<0\))
    • \(\theta_1\) equals the one-lag autocorrelation
    • Galton’s regression to the mean as a stationary AR(1)
    • \(\mathbb{E}[X_t|X_{t-1}]\) as a weighted average; mean-reversion intuition and applications
  • T7: Spurious Correlation
    • “Nonsense” correlations vs. spurious correlation (formal sense)
    • Famous nonsense correlations (tylervigen.com); Hendry (1980), Yule (1926)
    • Spurious correlation between two independent random walks
    • Simulation: independent \(N(0,1)\) random walks can show \(\rho=0.71\); sample correlation widely spread over \([-1,1]\), with \(|r| > 0.5\) about 40% of the time
    • “Correlation does not imply causation,” acute for non-stationary series; differencing/log remedies

Slide corrections

The slides in the lecture videos for this class contain the following errors.

  • T2, slide 4, writes the annual (fourth) difference as \(\Delta X_t = X_t - X_{t-4}\), the same symbol as the first difference \(\Delta X_t = X_t - X_{t-1}\) defined just above. Write the fourth difference as \(\Delta_4 X_t = X_t - X_{t-4}\) to keep the two apart.
  • T5, slide 8, says that the three random walks “are completely independent from each other, yet correlated”. Independent series have zero population correlation. The point of the example is that their sample correlation is often large, which is the spurious correlation discussed in T7.
  • T6, slide 7, ends its derivation of \(Var(X_t) = \sigma^2/(1 - \theta_1^2)\) with “So the mean is not a function of time”. It should say “the variance”.
  • T7, slide 9, says that “the absolute value of the correlation coefficient is (roughly) uniform over \([-1, +1]\)”, so that “nearly half the time” independent series show a correlation stronger than 0.5. An absolute value cannot be negative. It is the sample correlation \(r\) itself that ranges over \([-1, +1]\). In simulations of independent random walks, \(|r|\) exceeds 0.5 about 40% of the time.

Quiz

Before class, complete the review questions for class 6. These will be the basis of the quiz at the start of class. There will also be some additional questions based on the material covered in class 5.

Flashcards

Download the Class 6 flashcard deck and study it alongside the review questions. The flashcards are fair game on the quiz. See the Flashcards page for setup and study tips.

Problem Set

After attending class, complete problem set number six.

Date, Time, and Location

Thursday 11 March (HT Week 8), 3:30-6pm, Lyttleton Room, LMH

Videos

For this class there are three sets of videos: (1) my introduction to causal inference for a general audience, (2) the lecture videos, which are scheduled for release on Canvas in HT Week 4 (week beginning Sunday 7 February):

  • C0: Introduction
    • Causal inference vs. statistical inference (distinct)
    • “Population first” approach
    • Roadmap: potential outcomes, selection bias, RCTs, internal/external validity, natural/quasi experiments, LATE
  • C1: Potential Outcomes
    • Neyman–Rubin potential-outcomes framework; outcome \(Y\), binary treatment \(D\)
    • Potential outcomes \(Y(0)\), \(Y(1)\); individual effect \(Y(1)-Y(0)\)
    • ATE \(=\mathbb{E}[Y(1)]-\mathbb{E}[Y(0)]\); ATT (conditioned on \(D=1\))
    • Switching equation \(Y=Y(0)(1-D)+Y(1)D\)
    • Fundamental problem of causal inference; the counterfactual
  • C2: Selection Bias
    • Observed comparison \(\mathbb{E}[Y(1)|D=1]\) vs. \(\mathbb{E}[Y(0)|D=0]\)
    • Fundamental equation: observed difference \(=\) ATT \(+\) selection bias
    • Selection bias defined; arises when treatment depends on potential outcomes
    • Worked examples (Oxford-vs-University-of-X earnings; Youth Training Scheme); positive vs. negative bias
    • Selection bias distinct from sample bias; not directly observable
  • C3: Randomized Controlled Trials
    • RCTs to avoid selection bias; \(D\perp\!\!\!\perp\{Y(0),Y(1)\}\)
    • Randomization balances characteristics on average
    • Independence from potential vs. realized outcomes
    • Mean-independence; selection-bias term driven to zero
    • Under randomization ATT \(=\) ATE \(=\) observed difference in means
  • C4: Internal and External Validity
    • Internal validity (credible for the sample) vs. external validity (extrapolates to the population)
    • Threats to internal validity: contamination, non-compliance
    • Hawthorne effect; placebo effect
    • External validity defined; internal does not imply external
    • Sample/population mismatch; enumerative vs. eliminative induction; spillovers; short durations/surrogate outcomes
  • C5: Natural and Quasi-experiments
    • Alternatives when RCTs are infeasible/unethical
    • Randomization as the identifying mechanism (independence of assignment from potential outcomes)
    • Natural experiments (nature-induced variation)
    • Quasi-experiments (institutions, laws, borders); Angrist & Lavy (1999) Maimonides’ rule
  • C6: Conditional Independence
    • Conditional Independence Assumption: \(D\perp\!\!\!\perp\{Y(0),Y(1)\}\mid X\)
    • Selection on observables (Angrist 1998 military-service example)
    • Conditional decomposition; CIA eliminates the conditional selection-bias term
    • Recovering ATE/ATT by averaging cell-specific effects (Law of Iterated Expectations)
    • CIA is untestable/contestable; common support and the curse of dimensionality
    • Simpson’s paradox; “bad controls” (conditioning on an outcome) and the gender-pay-gap decomposition

and (3) some additional videos of mine that complement the lecture videos:

Slide corrections

The slides in the lecture videos for this class contain the following errors.

  • C6, slide 15, gives \(100 \times 0.2^5 = 0.035\). The correct value is 0.032.

Further reading (optional)

These posts from my blog go beyond the material in the lecture videos.

Quiz

Before class, complete the review questions for class 7. These will be the basis of the quiz at the start of class. There will also be some additional questions based on the material covered in class 6.

Flashcards

Download the Class 7 flashcard deck and study it alongside the review questions. The flashcards are fair game on the quiz. See the Flashcards page for setup and study tips.

Problem Set

After attending class, complete problem set number seven.